Superposition and mimicking theorems for conditional McKean–Vlasov equations

We consider conditional McKean–Vlasov stochastic differential equations (SDEs), as the ones arising in large-system limit of mean field games and particle systems with interactions when common noise is present. The time-marginals solutions to these SDEs are governed by non-linear partial (SPDEs) second order, whereas their laws satisfy Fokker–Planck on space probability measures. Our paper establishes two superposition principles: first asserts that any solution SPDE can be lifted a SDE, guarantees equation measures SPDE. use results obtain mimicking theorem which shows an Itô process emulated those SDE Markovian coefficients. This yields, particular, tool for converting open-loop controls into context controlled dynamics.